5A Effects of receptor clustering and rebinding on intracellular
signaling
In the analysis of bacterial chemotaxis, we showed how receptor clustering can amplify a biochemical signal through cooperative effects (see Sect. 5.3.1). Here we explore another consequence of membrane protein clustering, namely, it can increase
the likelihood of rapid rebinding of a ligand. In mammalian cells, clustering of receptors may by facilitated by lipid rafts, which are membrane microdomains rich in
cholesterol and having diameters in the range 25-200 nm [7, 9]. One suggested role
of lipid rafts is that they serve as mediators for several growth factors by localizing
clusters of their corresponding receptors. (Growth factors acts as triggers for many
cellular processes by binding to the extracellular domain of the receptors.) For example, in vitro experiments have shown that disruption of lipid rafts significantly
effects the dissociation of fibroblast growth factor-2 (FGF-2) from heparin sulfate
proteoglycans (HSPG), which is a co-receptor [5, 2].
Stochastic model of ligand rebinding
We begin by describing a self-consistent stochastic mean-field model of ligand rebinding due to Gopalakrishnan et al. [3]. Consider a homogeneous distribution of
receptors on a two-dimensional planar surface with mean surface density R0 per unit
area. Let R(t) denote the density of receptors bound to a ligand at time t, with
dR
= −k− R(t) + k+ ρ(t)[R0 − R(t)],
dt
(5A.1)
where k± are the intrinsic association and dissociation rates, and ρ(t) is the ligand density in a neighborhood of the surface (a boundary layer of width ∆ that is
comparable to the size of a ligand). Suppose that the initial density of ligand bound
receptors is R(0) = R∗ R0 and the initial density of ligand in the bulk is zero. It
follows that the only contribution to ρ(t) 6= 0 when t > 0 is from ligands released
from bound receptors at earlier times 0 < τ < t. Let G(r, ∆ ,t) denote the probability
density (per unit volume) that a ligand, which started in the boundary layer at time
t = 0, is in contact with the surface at time t having shifted by r ∈ R2 in the plane,
with possible multiple visits to the semi-absorbing surface at intermediate times but
without binding to a receptor. Since ρ(t) is independent of position in the plane, we
have
Z
Z t
ρ(t) = k− R(τ)
G(r, ∆ ,t − τ)dr dτ.
(5A.2)
0
R2
with k− dτ the probability that a ligand is released from a bound receptor in the
interval [τ, τ + dτ].
The next step is to calculate the Green’s function G(r, ∆ ,t). Let q(r, ∆ ,t) denote
the probability density (per unit volume) that a ligand released at time t = 0 first
2
returns to the surface at time t after shifting by r ∈ R2 in the plane. The Green’s
function satisfies the integral equation
Z t Z
0
0
0 dτ
. (5A.3)
G(r, ∆ ,t) = q(r, ∆ ,t) + γ
∆
q(r , ∆ , τ)G(r − r , ∆ ,t − τ)dr
δ
0
R2
Here δ is the time interval for which a ligand resides in a volume element ∆ 3 , before
diffusing way into the bulk. The first term on the right-hand side is the probability
that a ligand arrives back at the surface for the first time at t, whereas iterating the
second term yields all contributions from multiple visits to the surcae (with binding
to a receptor). The factor γ gives the probability of non-binding upon contact with
the surface and takes the form γ = 1 − k+ R0 δ /∆ . (If R∗ were comparable to R0 then
we would have to replace R0 by the density of unbound receptors.) It turns out that
one does not have to calculate q explicitly [4]. Let G0 (r, ∆ , r) denote the Green’s
function in the case γ = 1, for which the planar surface is purely reflecting, and
Z t Z
0
0
0 dτ
. (5A.4)
G0 (r, ∆ ,t) = q(r, ∆ ,t) +
∆
q(r , ∆ , τ)G0 (r − r , ∆ ,t − τ)dr
δ
0
R2
We now observe that G0 is the fundamental solution of the 3D diffusion equation in
the domain (r, z), r ∈ R2 , z > 0 with a reflecting boundary at z = 0, which is wellknown. We can then express G in terms of G0 by Fourier transforming equations
(5A.3) and (5A.4) with respect to r and Laplace transforming with respect to t:
b ∆ , s) = qb(k, ∆ , s) + γλ G(k,
b ∆ , s)b
G(k,
q(k, ∆ , s)
δ
and
b0 (k, ∆ , s) = qb(k, ∆ , s) + ∆ G
b0 (k, ∆ , s)b
G
q(k, ∆ , s).
δ
It follows that
b ∆ , s) =
G(k,
b0 (k, ∆ , s)
G
γ∆ b
1+
G(k, ∆ , s) ,
b0 (k, ∆ , s)
δ
1 + (∆ /δ )G
which can be rearranged to yield
b ∆ , s) =
G(k,
b0 (k, ∆ , s)
G
,
b0 (k, ∆ , s)
1 + k+ R0 G
(5A.5)
where we have used (1 − γ)(∆ /δ ) = k+ R0 .
Next, let p(t) = R(t)/R0 denote the fraction of bound receptors. Laplace transforming equations (5A.1) and (5A.2), we find that
(k− + s) pb(s) − p(0) = k+ ρb(s)
and
3
b ∆ , s).
ρb(s) = k− R0 pb(s)G(0,
Hence
p(0)
b ∆ , s).
, Σ (s) = k+ R0 G(0,
s + k− (1 − Σ (s))
√
b0 (0, ∆ , s) = 1/ Ds so that
It can be shown that G
pb(s) =
k+ R0
Σ (s) = √
.
Ds + k+ R0
(5A.6)
(5A.7)
and, consequently,
pb(s) =
p(0)
√
.
Ds
s + k− √
Ds + k+ R0
(5A.8)
Equation (5A.8) may be used to extract the short-time and long-time behavior of the
bound receptor fraction p(t). At short times (large s), we have pb(s) ≈ p(0)/(s + k− ),
which corresponds to exponential decay at the intrinsic rate k− , that is, rebinding
has no effect.√On the other hand, in the late time regime (small s), we have pb(s) ≈
p(0)/[s + k− Ds/(k+ R0 ]) which exhibits non-exponential behavior in time. More
explicitly
p(t) ∼ p(0)e−k− t ,
for t √
p(t) ∼ p(0)ect erfc( ct),
D
,
(k+ R0 )2
for t D
,
(k+ R0 )2
(5A.9a)
(5A.9b)
where c = D(k− /k+ R0 )2 .
Extension to receptor clusters
Gopalakrishnan et al. [3] used the above mean-field formalism to study the effects
of receptor clustering on ligand rebinding. Suppose that we focus at a point on the
surface within a cluster, which has increased receptor density R1 compared to the
background. The density of free ligand close to the surface at the given point is now
given by
Z t
ρ(t) = k−
Z
p(τ)
0
R2
R0 (|r|)G(r, ∆ ,t − τ)dr dτ.
(5A.10)
Here
R0 (r) = R1 + (R0 − R1 )Θ (r − ξ ).
That is, if the distance between the initial and final positions of a ligand in the plane
is smaller than the size of a cluster ξ , then the receptor density is R1 , whereas if the
ligand travels lateral distance greater than ξ then the receptor density is R0 , R0 < R1 .
4
For the sake of illustration, suppose that the receptor distribution consists of dense
isolated clusters so that R0 ≈ 0 and R1 is large. Then
ρ(t) ≈ k− R1
Z t
0
h
i
e0 (0, ∆ ,t − τ) 1 − e−ξ 2 /4D(t−τ) dτ,
p(τ)G
(5A.11)
e ∆ ,t) is the inverse Laplace transform of G(0,
b ∆ , s) with R0 → R1 :
where G(0,
p
b ∆ ,t) = √ 1 − k+ R1 e(k+ R1 )2 t/D erfc(k+ R1 t/D).
G(0,
D
πDt
(5A.12)
2
The factor 1 − e−ξ /4D(t−τ) arises from integrating the 3D Green’s function over a
disc of radius ξ in the plane. The Laplace transform of the bound receptor fraction
still has the form (5A.6) with
Z ∞
h
i
b ∆ ,t) 1 − e−ξ 2 /4Dt dt.
(5A.13)
Σ (s) = k+ R1
e−st G(0,
0
Performing an asymptotic analysis of pb(s) in the limit s → 0 shows that at large
times t ξ 2 /D and densely packed clusters (large R1 ) [3]
p(t) ∼ p(0)e−k− ξ0 t/2ξ ,
ξ ξ0 ≡
2D
.
k+ R1
(5A.14)
Thus, in the case of dense isolated clusters of receptors, the effective dissociation
rate of ligands at sufficiently long time-scales is reduced by a factor that is inversely
proportional to the size of the cluster. Moreover, ξ0 sets the critical cluster size below
which clustering has negligible influence.
It is interesting to relate the above result to a very different approach to determining an effective dissociation, which is rate based on the theory of Berg and Purcell
[1], see Sect. 2.4. Recall that for a spherical cell of radius a, with N receptors on its
surface and intrinsic ligand binding rate k+ , the net ligand flux (in the absence of
dissociation) is J = b
k+ ρ0 , where ρ0 is the far-field concentration of ligand and
4πaDNk+
b
k+ =
.
Nk+ + 4πaD
(5A.15)
Comparison with the diffusion-limited result in the limit k+ → ∞, suggests that the
effective absorption probability of a ligand in contact with the cell surface is
γ=
Nk+
.
Nk+ + 4πaD
(5A.16)
It follows that the corresponding probability of non-absorption is 1 − γ, indicating
that the possible effect of rebinding can be accounted for by taking the effective
dissociation rate to be [8]
5
b
k− = k− (1 − γ) = k−
4πaD
.
Nk+ + 4πaD
(5A.17)
Now imagine that a receptor cluster of size ξ is sufficiently dense that it effectively
acts like an absorbing disc of radius r. Assuming Nk+ 4πrD, we have
4k− D
4πrD
b
,
=
k− ≈ k−
Nk+
k+ rR1
after relating the receptor density within the cluster to the total number of receptors according to R1 = N/(πr2 ). This is consistent with the effective decay rate in
equation (5A.14) under the identification ξ = r/4.
The issue of protein clustering not only applies to receptors detecting extracellular signals (Sec. 5A), but also membrane-associated enzymes that play an important
role in activating intracellular signaling pathways. A canonical motif is a signaling
molecule in the cytoplasm being chemically modified by two antagonistic enzymes,
with an activating enzyme such as a kinase located in the cell membrane and a
deactivating enzyme such as a phosphatase distributed in the cytoplasm (see Sec.
9.1). The effectiveness of enzyme clustering on the substrate response depends on
the number of substrate sites that have to be activated. If only one site of the cytoplasmic molecule has to be activated, then enzyme clustering reduces the response,
since the effective target size presented to the substrate molecules is smaller. On
the hand, if more than one site has to be activated, then clustering can enhance the
response due to the rapid rebinding property of protein clusters [6]. This is shown
schematically in Fig. 5A.1.
Fig. 5A.1: (A) Clustering of enzyme molecules reduces their effective target size for substrate
molecules in the bulk, which decreases substrate activation. (B) On the other hand, clustering also
enhances rapid substrate rebinding, which increases substrate activation if multisite modification
is required.
Supplementary references
1. Berg, H. C., Purcell, E. M.: Physics of chemoreception. Biophys. J. 20, 93-219 (1977).
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2. Chu, C. L., J. A. Buczek-Thomas, and M. A. Nugent. Heparan sulfate proteoglycans modulate fibroblast growth factor-2 binding through a lipid-raft-mediated mechanism. Biochem. J.
379:331-341 (2004).
3. Gopalakrishnan, M., Forsten-Williams, K., Nugent, M. A., Taubery, U. C. Effects of receptor
clustering on ligand dissociation kinetics: theory and simulations. Biophys. J. 89, 3686-3700
(2005).
4. Ghosh, S., Gopalakrishnan, M., Forsten-Williams, K.: Self-consistent theory of reversible ligand
binding to a spherical cell. Phys. Biol. 4 344-345 (2007).
5. Kramer, K. L., and H. J. Yost. 2003. Heparan sulfate core proteins in cell-cell signaling. Annu.
Rev. Genet. 37:461-484.
6. Mugler, A., Bailey, A.G., Takahashi, K., ten Wolde, P.R., 2012. Membrane clustering and the
role of rebinding in biochemical signaling. Biophys. J. 102, 1069-1078.
7. Munro, S. 2003. Lipid rafts: elusive or illusive? Cell. 115:377-388.
8. Shoup, D., Szabo, A.: Role of diffusion in ligand binding to macromolecules and cell-bound
receptors Biophys. J. 40, 33-39 (1982)
9. Simons, K., and W. L. C. Vaz. 2004. Model systems, lipid rafts and cell membranes. Annu. Rev.
Biophys. Biomol. Struct. 33:269-295
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